Just a note about vacuum radiation and similarities
When we solve for Casimir energy, we first quantize a field in free space and ask what frequency modes it can support. The answer is all of them because free space is isotropic and homogeneous. We then put in boundaries, either conducting or dielectric, and then as what frequency modes the space between the boundaries can support. The answer will in general be different because the boundaries change what frequency modes can be supported. For example, with parallel conductors, only the modes that can fit multiple half-wavelengths perfectly between the conductors will be supported.
It's wasn't immediately apparent to me, but Fulling-Unruh-Hawking radiation is a very similar process. The first step is identical: solve for the available frequency modes in free space. The second step is to solve for the modes in a new space-time coordinate system. A Bogoliubov transform will tell you if all the same modes are supported in both systems. If the transform's result is one, then the coordinate transform was unitary and corresponded to a simple rotation or translation. If the transform does not return a one, then the transformation 'stretched' the space-time so that it would no longer support the same frequency modes. An example of this was pointed by Fulling to be motion under constant acceleration which was named hyperbolic motion by Rindler.